Zero intersection graph of annihilator ideals of modules

Year 2025, Volume: 54 Issue: 6, 2182 - 2194, 30.12.2025

Seçil Çeken , Osama A. Naji , Ünsal Tekir , Suat Koç

https://doi.org/10.15672/hujms.1485903

https://izlik.org/JA67ZX39BE

Abstract

This paper aims to associate a new graph to nonzero unital modules over commutative rings. Let $R$ be a commutative ring having a nonzero identity and $M$ be a nonzero unital $R$-module. The zero intersection graph of annihilator ideals of $R$-module $M$, denoted by $\mathfrak{C}_{R}(M)$, is a simple (undirected) graph whose vertex set $M^{\star}=M-\{0\},\ $and two distinct vertices $m$ and $m^{\prime}$ are adjacent if $ann_{R}(m)\cap ann_{R}(m^{\prime})=(0).$\ We investigate the conditions under which $\mathfrak{C}_{R}(M)$ is a star graph, bipartite graph, complete graph, edgeless graph. Furthermore, we characterize certain classes of modules and rings such as torsion-free modules, torsion modules, semisimple modules, quasi-regular rings, and modules satisfying Property $T$ in terms of their graphical properties.

Keywords

property A , property T , torsion-free module , quasi regular ring , weak quasi regular module

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